Jaffard, Stéphane On lacunary wavelet series. (English) Zbl 1063.60053 Ann. Appl. Probab. 10, No. 1, 313-329 (2000). Summary: We prove that the Hölder singularities of random lacunary wavelet series are chirps located on random fractal sets. We determine the Hausdorff dimensions of these singularities, and the a.e. modulus of continuity of the series. Lacunary wavelet series thus turn out to be a new example of multifractal functions. Cited in 2 ReviewsCited in 43 Documents MSC: 60G17 Sample path properties 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable 28A80 Fractals 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems Keywords:Wavelet bases; Hausdorff dimensions; chirps; Hölder regularity; modulus of continuity PDFBibTeX XMLCite \textit{S. Jaffard}, Ann. Appl. Probab. 10, No. 1, 313--329 (2000; Zbl 1063.60053) Full Text: DOI References: [1] Arenedo, A., Bacry, E. and Muzy, J.-F. (1995). The thermodynamics of fractals revisited with wavelets. Phys. A 213 232-275. · Zbl 1064.80002 [2] Benassi, A., Jaffard, S. and Roux, D. (1997). Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13 19-90. · Zbl 0880.60053 [3] DeVore, R., Jawerth, B. and Popov, V. (1992). Compression of wavelet decompositions Amer. J. Math. 114 737-795. JSTOR: · Zbl 0764.41024 [4] DeVore, R. and Lucier, B. (1990). High order regularityfor conservation laws. Indiana Univ. Math. J. 39 413-430. · Zbl 0714.35018 [5] DeVore, R. and Lucier, B. (1992). Fast wavelet techniques for near-optimal image processing. Proceedings IEEE Mil. Comm. Conf. [6] Donoho, D. (1995). De-Noising via soft-thresholding. IEEE Trans. Inform. Theory 41 613- 627. · Zbl 0820.62002 [7] Donoho, D., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: asymptopia? J. Roy. Statist. Soc. Ser. B 57 301-369. JSTOR: · Zbl 0827.62035 [8] Falconer, I. (1990). Fractal Geometry. Wiley, New York. · Zbl 0689.28003 [9] Frisch, U. and Parisi, G. (1985). Fullydeveloped turbulence and intermittency. Proc. Internat. Summer School of Phys. Enrico Fermi 84-88. North-Holland, Amsterdom. [10] Guiheneuf, B., Jaffard, S. and Lévy-Véhel, J. (1998). Two results concerning chirps and 2-microlocal exponents prescription. Appl. Comput. Harmon. Anal. 5 487-492. · Zbl 0930.42022 [11] Houdré, C. (1994). Wavelets, probabilityand statistics: some bridges. In Wavelets: Mathematics and Applications (J. J. Benedetto and M. Frazier, eds.) 365-398. CRC Press, Boca Raton, FL. · Zbl 0886.60045 [12] Jaffard, S. (1991). Pointwise smoothness, two-microlocalization and wavelet coefficients. Publ. Mat. 35 155-168. · Zbl 0760.42016 [13] Jaffard, S. (1997). Multifractal formalism for functions. SIAM J. Math. Anal. 28 944-998. · Zbl 0876.42021 [14] Jaffard, S. and Meyer, Y. (1996). Wavelet methods for pointwise regularityand local oscillations of functions. Mem. Amer. Math. Soc. 123. · Zbl 0873.42019 [15] Kahane, J.-P. (1985). Some Random Series of Functions, 2nd ed. Cambridge Univ. Press. · Zbl 0571.60002 [16] Lemarié, P.-G. and Meyer, Y. (1986). Ondelettes et bases hilbertiennes. Rev. Mat. Iberoamericana 1. · Zbl 0657.42028 [17] Meyer, Y. (1990). Ondelettes et Opérateurs. Hermann, Paris. · Zbl 0745.42011 [18] Meyer, Y. and Xu, H. (1997). Wavelet analysis and chirps. Appl. Comput. Harmon. Anal. 4 366-379. · Zbl 0960.94006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.